The unique solution of the equation \(\partial _\phi \omega _\phi + L_{V_\phi }\omega _\phi = 0\)  is \(\omega _\phi = (F_{\phi ,0})_*\omega _0\) , where \((F_{\phi ,\phi _0})_*\)  denotes the pushforward[1]} along \(F_{\phi ,\phi _0}\) . Setting \(\phi = 2\pi \) , we therefore obtain the fundamental result \((F_{2\pi ,0})_*\omega _0 = \Phi _*\omega _0 = \omega _0\) , or equivalently \(\Phi ^*\omega _0 = \omega _0\) , which says that the mapping \(\Phi :D^2\rightarrow D^2\)  preserves the 2-form \(\omega _0\) . If the component functions of \(F_{2\pi ,0}\)  are denoted \(F_{2\pi ,0} = (\overline{x},\overline{y})\)  then the condition \(\Phi ^*\omega _0 = \omega _0\)  may also be written
\(B^\phi (\overline{x},\overline{y},0)\,\mathcal {J}(\overline{x},\overline{y},0)\,d\overline{x}\wedge d\overline{y} = B^\phi (x,y,0)\,\mathcal {J}(x,y,0)\,dx\wedge dy.\) 
